A counterexample to sparse removal

Abstract

a b s t r a c tThe Turán number of a graph H, denoted ex(n, H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi (2011) conjectured that if H is any graph and ex(n, H) = O(n α ) where α > 1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(n α ) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemerédi (1978). Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.

Generalized Turán problems for complete bipartite graphs

“…Lazebnik and Verstraëte [22] studied r-uniform hypergraphs without Berge cycles of length at most 4, and proved the bound Ω(n 4/3 ). Timmons and Verstraëte [30] mention that this can be improved to n 3 2 −o(1) using a construction of Ruzsa [26], thus we have the same lower bound for ex(n, K a,b , K 2,t ) if 2 < a ≤ b < t. In general, the above proposition gives a superlinear lower bound Ω((n log n) 1+1/(2s+1) ) in conjunction with a result of Shangguan and Tamo [28]. They showed that for any k, there exists a k-uniform hypergraph without Berge cycles of length at most ℓ that has Ω((n log n) 1+1/(ℓ+1) ) hyperedges.…”

On $r$-Uniform Linear Hypergraphs with no Berge-$K_{2,t}$

“…The idea is that a copy of some small graph in our construction corresponds to a nontrivial solution to some system of equations over F q . Variations of these lemmas have appeared in [15].…”

“…where each of the edges of this K 2,2 have color m s . Using (15) as our condition for adjacency in H(V i , V j ), we have…”

“…Section 3.1 contains algebraic lemmas which are required for our construction. Section 3.2 gives the construction which is a generalization of the one found in [15] and is based on a construction Allen, Keevash, Sudakov, and Verstraëte (see Theorem 1.6 [1]). Using the counting argument of [13] we can prove an upper bound on the number of edges in a {C 2 , C 3 , K 2,t+1 }-free r-uniform hypergraph.…”

Relative Turán Numbers for Hypergraph Cycles

“…For , r ≥ 3, Győri and Lemons [13] proved ex(n, C r ) = O(n 1+ 1 /2 ), so up to the o(1) term, ( * ) would be best possible. This conjecture is known to hold for = 3 and all r due to work of Ruzsa and Szemeredi [24] and Erdős, Frankl, and Rödl [5], and for = 4 and all r due to Lazebnik and the second author [16] and Timmons and the second author [27]. We emphasize that for r ≥ 3, it is known that the o(1) term in ( * ) is necessary in general, see Ruzsa and Szemeredi [24] and Conlon, Fox, Sudakov, and Zhao [3].…”